Basic Graphing on TI 83 / 84 A graphing calculator can, of course, graph but only from an equation in function form. That means each equation must be solved for "y". The first activity is to practice solving equations for y. Remember there can be one y and it must be isolated. Some equations must be rewritten as two equations in order for them to be in function form. Example: To solve y 2 +5 = x for y isolate the y term y 2 = x 5 use the Principle of Powers to remove 'square' on y, gives both + and solutions y = ± x 5 This must be written as two separate equations for y1 = x 5 graphing y2 = x 5 GRAPHING keys The keys that access the graphing functions are all located in a row just below the screen. Both the keys and the 2 nd functions are important in using this feature. FORMAT Before you begin graphing check the FORMAT settings. These settings affect the way the graph is displayed. (If you ever get an odd looking result this is a good place to start in trying to correct it.) Usually the first entry on each on each line should be highlighted. To change a setting move the cursor to the desired setting and press ENTER. The selections are pretty self-explanatory. To get out of the FORMAT screen choose the next item that you want or QUIT will take you back to the home screen, the changes you made will be retained. Y = : Entering a function: To enter the function you want to graph, select Y =. You see a list of y equations that can be or are entered and, across the top of the screen, a list of possible PLOTS Type in your function for y1. If you have more than one function use the cursor down key and type in y2. Always use the "x" from the X,T,θ,n key for your variable in graphing. IMPORTANT: The plots at the top of the screen should be turned off unless you are working with a plot of a statistics problem. A plot is on if it is highlighted Plot1. To turn on or off, position the cursor on the Plot and press ENTER. Again for most graphing all the plots should be off. Getting rid of an Equation: If there are already equations entered in the calculator, you can remove them by using the CLEAR key with the cursor is on the appropriate line. Selecting/Deselecting an equation: When the Equal signs show as highlighted, =, the equation is selected and the equation will be graphed. The equals sign automatically highlights when you enter an equation. To choose not to graph a function you must "turn off'" the selection. Use the arrow keys to position the cursor on the equals sign and press ENTER. Use the same procedure to "turn back on" the selection of an equation.
Setting the WINDOW: After you have entered your equation, you have to tell your calculator what range of values of x and y you want displayed on the screen. Select WINDOW. You can now enter values to set the part of the graph you want to see You now see WINDOW xmin = 10 xmax = 10 xscl = 1 ymin = 10 ymax = 10 yscl = 1 Xres = 1 These values are the default settings and if your calculator shows different values it just means that they were changed for the last graph. You need to set the maximum and minimum values of x and y so that the screen will show the part of the graph you are interested in looking at. If you do not see the graph of your equation, then you are not viewing an area where the function passes through. Remember that when entering negative values in the WINDOW settings you must use the ( ) key for a negative number and not the (minus) key. Also be sure that your minimum value is less than your maximum value. This is a common error and will give you an error message when you try to graph. The picture of the screen at the right is to show you what the MINs and MAXs do for you The SCLs set the scale marking on the x and y-axes. If you have too many "tick marks" the axis will appear fat. This may happen when you use the ZOOM features. XRes determines how frequently the calculator finds points as it plays connect the dot to graph the equation. The number is in pixels You will want to become comfortable changing the view window to match your function. Remember that the xmin value must be less than the xmax and the likewise for the y values. Follow the example to get a hang of what these values do. Example: Graph 2y 6x = 10 Solve for y y = 3x + 5 Enter the equation Y = Set Standard window 3 [X,T,θ,n] + 5 for y1 y1 = 3 x + 5 Select WINDOW check for the values shown above if they are different, position the cursor and type in the correct values Standard WINDOW settings Now graph GRAPH Shows graph of line with a y- intercept at (0, 5) Look at a different part of the graph 5 < x < 50 by changing you WINDOW values WINDOW position cursor and type in values to match Xmin = 5 Xmax = 50 Xscl = 5 (hatch mark every 5) Ymin = 10 Ymax = 145 Yscl = 10 Now graph again GRAPH graph appears no axis on screen If you want to see the very bottom of the graph screen press CLEAR once. press GRAPH again to get the menu line back
The ZOOM menu contains some useful tools ZStandard: Returns the WINDOW values to the default settings (see these settings above). This is a good place to start with most of the graphs you will work with in this class. In the previous example you would just choose ZStandard instead of selecting WINDOW and GRAPH. ZSquare: The calculator does not give a "square" view of the graph unless the WINDOW values are set appropriately. Usually the x-axis "stretched" when compared to the y-axis. Use the "Zoom Square" option to get rid of the distortion. ZPrevious: Found in the ZOOM MEMORY sub-menu, will returns you to the previous WINDOW settings. Example: graph y 2 + x 2 = 16 Solve the equation for y Enter these as y1 and y2 and graph in the standard Window view a square picture of the graph of the circle y 2 + x 2 = 16 y1 = y2 = 16 x 2 16 x Y = type in equations notice the parenthesis! ZOOM ZSTD The current picture looks like and egg instead of a circle, so choose ZOOM ZSQR and see the difference in the graph. 2 Notice you must have two separate equations! y1 = (16 x 2 ) y2 = (16 x 2 ) Look at the change in the WINDOW values Return to the previous screen WINDOW ZOOM MEMORY ZPREVIOUS Check the values and compare to standard settings or to always get back to standard settings use [ZOOM] ZStandard More ZOOM features ZoomFit: Sometimes when you graph a function you will get a blank screen. First make sure the function is selected (highlighted = sign). If that is fine, then you are probably just looking at the wrong part of the graph picture, you will need to adjust the [WINDOW]. Start by checking that your function is defined between the xmin and xmax values you set. Then from the ZOOM menu choose ZoomFit. The calculator will think and then adjust the ymax and ymin to show the graph in the window. You can now make additional adjustments to the [WINDOW] values to suit and change SCL to a reasonable value to remove a fat axis. Example: Graph y = x + 25, adjust window to see the graph keystroke display In Graph Y = type in equation for y1 x + 25 y1 = x + 25 graph in standard window ZOOM ZStandard no graph line visible find the graph ZOOM ZoomFit now graph is shown
Zoom Out, Zoom In and BOX will zoom on the part of the graph you are interested in. For all of them you must tell the calculator where you want to ZOOM. In Zoom Out and Zoom In set the cursor to the center of where you are interested in using the arrow keys and press ENTER. For BOX you must set the diagonal corners of box with the cursor and ENTER after each. Example: on the same graph take a close look at the y intercept and then zoom out to find the x intercept and use ZBOX to look at x intercept keystroke display Zoom in on the y intercept ZOOM Zoom In cursor blinks at center of screen move cursor around for practice when cursor as a the center of the new desired picture, press ENTER again Zoomed in screen go back to previous screen ZOOM MEMORY ZPrevious Previous picture Zoom out with the lower left hand corner as center of next screen ZOOM Zoom Out move cursor to lower left and corner and ENTER can now see a x intercept use a zoom box to look at x intercept make a box around intercept ZOOM ZBox move cursor to a position up and right of the intercept and ENTER use arrow keys to create box by moving to opposite diagonal corner ENTER cursor changes to little box instead of + Zoomed in screen Other Graphing Tools TRACE: When you select TRACE and use the left and right arrows, a cursor moves along the graph and the values of x and y are displayed at the bottom of the screen. Also in TRACE, you can enter any x value that is shown on the current screen followed by ENTER and the calculator will move the cursor there and return the x and y coordinates of the point. If the x value is not on your current screen the calculator will return an error, go to WINDOW and adjust you xmin and xmax to include the number you are interested in. Example: On the same graph, Trace to approximate the x intercept and use TRACE to find the value of y when x is 2.63 keystroke Display Get into TRACE TRACE equation in upper right hand corner and x and y values at bottom of screen move along graph Right or left arrow until close to x intercept closest I could get was x = -25.04 and y = -.047 We know of course that it To find the value of y when x is 2.63 you must first adjust ot window to include this value of x go back to TRACE and ask for x = 2.63 in WINDOW set xmin to 10 and xmax to 10, then move to ZOOMand Zoom Fit to see graph should be x = -25 and y = 0 adjusted graph TRACE 2.63 ENTER x = 2.63 y = 27.63 Now readjust the window and use TRACE to find the y intercept
Assignment Check you graphing ability with these exercises 1) Graph the quadratic function y = 3x 2 +2x 5, in the standard window, Sketch the graph you see. Use Trace or Eval to determine y when x = 5.72 y = 104.5952 2) Enter a second equation y = -2x 3 + 6x + 5. Keep the first equation but un-highlight the equals sign so only the second equation graphs. Sketch the graph as you see it on the screen. Zoom in on the x intercept, is the x value at the x intercept greater or less than 2? x intercept greater than 2 3) Delete or clear both of your equation and enter y =5x 2 6x + 15 for y1. Graph in the standard window and you will not see a graph. Use Zoom Fit to find the graph, sketch what you see. see a parabola once the Zoom is adjusted with heavy y axis 4) With the same graph (y =5x 2 6x + 15) reset the xmax and xmin values for your window to show the part of the graph between an x of -113 and -115. Use Zoom Fit to find it, Sketch the graph and find the value of y when x is -113.25 (use TRACE or EVAL) y = 64822.3125 5) Graph the equation y = x 2 + 3x + 8. Adjust your Window to show the graph for x's from 20 to 20 and y's from 5 to 350, with x and y SCL values at 1, Sketch what you see. Now change the y SCL to 50. Graph again. What has changed? y axis scale changed so that now you see tick marks instead of a fat axis
More Graphing on TI 83 Finding intersections; X intercepts; and Maximum and Minimums Finding the intersection of two graphs Enter and graph both equations and then go to ISECT, found in the CALC Menu. On the graph, position the cursor on one of the lines you are interested in using, the up or down arrows, and press [ENTER]. Now position the cursor on the other graph line you want and press [ENTER] again. Follow the directions for your calculator type from here. The Calculator will find the coordinate of the intersection and display them at the bottom of the screen. Note that the intersection must be in the view rectangle for this to work. You will need to find the coordinates of each intersection separately. The system does NOT have to be linear for this approach. Example: Find the solution to this system of equations 2x + 3y = 6 5x 8y = 11 First solve each equation for y and enter in [GRAPH] y= then type in the equations y1 = ( 2/3) x + 2 y2 = (5/8) x 11/8 graph in the standard window ZOOM ZSTD you should see two lines that intersect go into the CALC menu Scroll down to find find ISECT ISECT cursor appears on graph of y1, and 1 appears in the upper right hand corner accept line y1 move close to intersection, then [ENTER] accept line y2 move close to intersection, then [ENTER] : you must also enter a guess use left or right arrow and position cursor close to intersection and [ENTER] cursor appears on y2 and a 2 appears in the upper right hand corner at bottom of the screen ISECT x = 2.612.. y =.25806 Check yourself on this feature 1. Graph y1 = 5x+3 in the standard window. Also graph y2 = 12. Adjust you window so that you can see both graphs. Sketch what you see. How does this combination of graphs relate to the inequality 5x+3>12? The x value at the intersection is the solution x = 1.8
Finding the x-intercepts using zero zero is also found in the CALC menu. Use your ZOOM features to get a good picture of the x intercept you are interested in, then choose zero from the CALC menu and the cursor will appear on the graph. The up and down arrows are used to move between graphs, if more than one is currently displayed. The left and right arrows move along a graph line. The calculator asks you for a left bound so position the cursor to the left of the intercept and press ENTER. Now the calculator asks the right bound so position the cursor to the right of the intercept and press ENTER again. Finally it asks for a guess, so position the cursor close to the intercept and press ENTER one last time and now the calculator will display the coordinates of the intercept. Example: Find the solutions to 3x 4 5x +8 = 0 Enter the equation to be Y= then type in equation y1 = 3x^4 5x + 8 graphed graph in the standard window ZOOM ZStandard graph is displayed you can see the x-intercepts so find the zeros CALC zero you see a cursor on the graph and Left Bound? Enter a left bound for where you want the calculator to look for an intercept Use the right or left arrow to position the cursor on the left side of the intercept you are The calculator puts a little arrow at the top of the screen to mark the x value and asks Enter a right bound Make a guess to find the other intercept press CALC again and start over looking at, and ENTER Use the right arrow to position the cursor to the right side of the intercept Use the left arrow to bring the cursor a little closer to the intercept, it does not have to be right on and ENTER CALC follow the same procedure for the other zero Right Bound? The calculator puts a little arrow to mark the right bound and asks Guess? Coordinates appear at bottom of screen. For positive intercept x = 1 y = 0 Coordinates appear at bottom of screen. for positive intercept x = 1.5087.. y = 1E 12 Check yourself on this feature 6) Find the solutions to this equation: 0 = 3x 4 6x 3 + 2x 2 + 3x 5. The solutions to a polynomial = 0 occur at the x intercepts of the graph. Notice that there are two intercepts, you must find each one separately. Adjust your window as necessary. x = -0.88887 and x = 1.5996
Finding Maximum and Minimums Polynomial functions may have changes of direction. The top or bottom point of the direction change is called the vertex. When considering the maximum or minimum of a polynomial over a specified region, the maximum or minimum will occur either at a vertex or at one of the endpoints. The calculator can find the maximum or minimum value of a function over a particular region. TI 83 In [CALC] menu find minimum or maximum. These functions are similar to the zero option in that you will need to enter a left bound, a right bound and a guess before the calculator will find the maximum or minimum value. It will consider the endpoints and the any vertex that occurs so be careful in setting the bounds. Example: Find the minimum value of y = x 3 5x 2 + 7x 9 between x = 1 and x = 3 Keystrokes Display Enter the function y1 = x ^ 3 5x 2 + 7x 9 graph in Standard window ZOOM ZStandard notice the dip between x = 1 and x = 3 enter CALC menu and find CALC minimum left bound? minimum Set the left bound position cursor using the arrow right bound? keys, ENTER Set the right bound position cursor using the arrow guess? keys, ENTER make a guess position cursor using the arrow keys, ENTER coordinates of the minimum x = 2.333.. y = 7.185185 Check yourself on this feature 1) Graph y = x 3 5x 2 + 2x 1. Adjust your window so that you can clearly see the local Maximum and Minimum, sketch you graph. Now use the FMAX (maximum on 83) or FMIN (minimum on 83) to approximate the x and y values at these local maximum and minimum points local max at (0.2137,.7911) local min at (3.1196, -13.061)